Anti-Compacts and Their Applications to Analogs of Lyapunov and Lebesgue Theorems in Frechét Spaces

Abstract

We introduce anti-compact sets (anti-compacts) in Frechét spaces. We thoroughly investigate the properties of anti-compacts and the scale of Banach spaces generated by anti-compacts. Special attention is paid to systems of anti-compact ellipsoids in Hilbert spaces. The existence of a system of anti-compacts is proved for any separable Frechét space E. Using the constructed theory, we obtain analogs of the Lyapunov theorem on the convexity and compactness of the range of vector measures in the class of separable Frechét spaces: We prove the convexity and compactness of the range of vector measure in a space \( {E}_{\overline{C}} \) generated by an anti-compact \( \overline{C} \). Also, the nondifferentiability problem with respect to the upper limit is investigated for the Pettis integral. We obtain differentiability conditions for the indefinite Pettis integrals in terms of the new weak integral boundedness and the σ-compact measurability. We prove an analog of the Lebesgue theorem on the differentiability of the indefinite Pettis integral for any strongly measurable integrand.

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 53, Proceedings of the Crimean Autumn Mathematical School-Symposium KROMSH-2013, 2014.